In Physics, and especially in Mechanics, it is crucial to be able to know where the object we are considering was (position) and when it was there (time). Thus, measuring time and position is fundamental and we need to say a few words about it.

Measuring time:
The motivation to measure time stems from wanting to answer two fundamental questions: ”When did this event occur?” and ”How long did this event last?”

To measure time, we need to know when we started to measure it, i.e. we need an origin for time. You can pick any moment during the motion and call it $t=0$ though you would typically choose a moment so that time is always positive. Usually, we like to choose $t=0$ at the beginning of the motion, but this does not mean that the object has to be at rest. Initially, your object could be at rest or moving around with some velocity, all that matters is that you have a clear origin of time to analyze your problem.

Measuring position:
To measure position, you will need what is called a coordinate system. You may think of it as a ruler that you lay in the direction of motion, whose origin you place at a specific point that you choose and that you use to measure position. You will see that, in general, motion does not take place in one dimension and it is customary to have at least two directions (usually labelled $x$ and $y$) along which you need to place a ruler. The two directions $x$ and $y$ are chosen to be orthogonal for convenience and are independent. The direct consequence of the axes being independent is that you can analyze motion in one direction and then in the other and obtain separate equations for each.

In addition, for a given axis ($x$ or $y$), one direction is chosen positive and the other negative. Typically, the directions ”to the right” and ”upward” are assigned a positive value while the directions ”to the left” and ”downward” are assigned a negative value. This is especially helpful to be able to tell if an object is moving to the left or the right.

For instance, in the example below the motion of the ball would be assigned positive quantities in the coordinate system $\left(x_1,y_1\right)$ since it defines ”to the right” as being the positive horizontal direction. In the $\left(x_2,y_2\right)$ coordinate system, the motion of the ball would be assigned negative quantities since it defines ”to the left” as being the positive horizontal direction.

Of course, the choice of making one direction positive over the other is arbitrary and you can decide to make ”to the left” and ”downward” positive or chose any combination as long as you state your choice explicitly by drawing the axes as is done in the figure above. It only matters that, once you choice is made, you be consistent and not change coordinate systems in the middle of a problem.

Finally, you may choose the origin of the coordinate system (i.e. the location where $x=y=0$) at any arbitrary point. In practice, however, we will customarily choose specific points consistently because they make the most sense. For example, it might be a good idea to choose the origin to be at the location where the motion begins. Over the course of solving problems, you will see that choosing the origin will become second nature.

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